Directional upper derivatives and the chain rule formula for locally Lipschitz functions on Banach spaces

Abstract

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An inverse function theorem converse

“…In this section we recall some notions of differentiability of Lipschitz functions between open subsets of a Banach spaces and a generalized chain rule from the work of Maleva and Preiss [4]. This material will be crucial to the derivation of our main results in the next section.…”

“…If the directional derivative of f at y in the direction v exists for all v ∈ Y , then the mapping from Y to Z sending v to its directional derivative is, by definition, the Gâteaux derivative (by some authors the mapping is also required to be a continuous linear map). [4] have given the following generalization of the one-sided directional derivative.…”

“…Helpful tools for the task at hand have been developed in the intense study of Lipschitz functions in the Banach space setting motivated by Rademacher's theorem concerning the existence of an abundance of points of differentiability of Lipschitz mappings in the setting of euclidean spaces. In particular we recall in the next Date: December, 2018. section a useful generalization of the chain rule by O. Maleva and D. Preiss [4]. In Sections 3 and 4 we present our main results and provide some follow-up illustrative material in Section 5.…”

“…section a useful generalization of the chain rule by O. Maleva and D. Preiss[4]. In Sections 3 and 4 we present our main results and provide some follow-up illustrative material in Section 5.…”

On the differentiability of Lipschitz functions with respect to measures in the Euclidean space

“…We begin this section with a definition that is related to the notion of derivative assignment introduced in [16]; note that Proposition 3.3 below-the key result in this section-can be viewed as a particular case of a general chain-rule for Lipschitz maps proved in that paper. (ii) Both D(f, x) and D * (f, x) are closed and nonempty, and viewed as multifunctions in the variable x are Borel measurable; 1 since these properties play only a minor role in this proof, we postpone the precise statement (Lemma 7.8) to Section 7.…”